Why and when did ancient scholars make the enormous effort to understand the principles and master the mathematics of foreign astral sciences? This work provides a detailed analysis of the invention, development and transmission of astronomy, astrology, astral religion, magic and medicine, cosmology and cosmography, astral mapping, geography and calendrics and their related mathematics and instrumentation in and between Mesopotamia, Egypt, the West Semitic areas, Greece and Rome, Iran, India and China. It considers the available textual evidence from the most ancient times to the seventh century CE. The author has worked the contributions of eight internationally renowned scholars into what amounts to a new history of the oldest sciences. The result is a challenging read for the layperson and a resource for the expert and includes an extensive index to the entire volume. It provides a new typology of cultural interactions and, by describing their socio-political backdrop, offers a cultural history of the region. In particular, astral science in the Hellenistic period west of the Tigris is completely re-evaluated and a new model of the interactions of Western and Indian and Iranian astral sciences is provided.